In computer science, a queue is a particular kind of abstract data type or collection in which the entities in the collection are kept in order and the principal (or only) operations on the collection are the addition of entities to the rear terminal position, known as enqueue, and removal of entities from the front terminal position, known as dequeue. This makes the queue a First-In-First-Out (FIFO) data structure. In a FIFO data structure, the first element added to the queue will be the first one to be removed. This is equivalent to the requirement that once a new element is added, all elements that were added before have to be removed before the new element can be removed. Often a peek or front operation is also entered, returning the value of the front element without dequeuing it. A queue is an example of a linear data structure, or more abstractly a sequential collection.

Queues provide services in computer science, transport, and operations research where various entities such as data, objects, persons, or events are stored and held to be processed later. In these contexts, the queue performs the function of a buffer.

Theoretically, one characteristic of a queue is that it does not have a specific capacity. Regardless of how many elements are already contained, a new element can always be added. It can also be empty, at which point removing an element will be impossible until a new element has been added again.

Fixed length arrays are limited in capacity, but it is not true that items need to be copied towards the head of the queue. The simple trick of turning the array into a closed circle and letting the head and tail drift around endlessly in that circle makes it unnecessary to ever move items stored in the array. If n is the size of the array, then computing indices modulo n will turn the array into a circle. This is still the conceptually simplest way to construct a queue in a high level language, but it does admittedly slow things down a little, because the array indices must be compared to zero and the array size, which is comparable to the time taken to check whether an array index is out of bounds, which some languages do, but this will certainly be the method of choice for a quick and dirty implementation, or for any high level language that does not have pointer syntax. The array size must be declared ahead of time, but some implementations simply double the declared array size when overflow occurs. Most modern languages with objects or pointers can implement or come with libraries for dynamic lists. Such data structures may have not specified fixed capacity limit besides memory constraints. Queue overflow results from trying to add an element onto a full queue and queue underflow happens when trying to remove an element from an empty queue.

A doubly linked list has O(1) insertion and deletion at both ends, so is a natural choice for queues.

A regular singly linked list only has efficient insertion and deletion at one end. However, a small modification—keeping a pointer to the last node in addition to the first one—will enable it to implement an efficient queue.

Queues may be implemented as a separate data type, or may be considered a special case of a double-ended queue (deque) and not implemented separately. For example, Perl and Ruby allow pushing and popping an array from both ends, so one can use push and unshift functions to enqueue and dequeue a list (or, in reverse, one can use shift and pop), although in some cases these operations are not efficient.

Queues can also be implemented as a purely functional data structure.[2] Two versions of the implementation exist. The first one, called real-time queue,[3] presented below, allows the queue to be persistent with operations in O(1) worst-case time, but requires lazy lists with memoization. The second one, with no lazy lists nor memoization is presented at the end of the sections. Its amortized time is O(1){\displaystyle O(1)} if the persistency is not used; but its worst-time complexity is O(n){\displaystyle O(n)} where n is the number of elements in the queue.

Let us recall that, for l{\displaystyle l} a list, |l|{\displaystyle |l|} denotes its length, that NIL represents an empty list and CONS⁡(h,t){\displaystyle \operatorname {CONS} (h,t)} represents the list whose head is h and whose tail is t.

The data structure used to implement our queues consists of three linked lists(f,r,s){\displaystyle (f,r,s)} where f is the front of the queue, r is the rear of the queue in reverse order. The invariant of the structure is that s is the rear of f without its |r|{\displaystyle |r|} first elements, that is |s|=|f|−|r|{\displaystyle |s|=|f|-|r|}. The tail of the queue (CONS⁡(x,f),r,s){\displaystyle (\operatorname {CONS} (x,f),r,s)} is then almost (f,r,s){\displaystyle (f,r,s)} and
inserting an element x to (f,r,s){\displaystyle (f,r,s)} is almost (f,CONS⁡(x,r),s){\displaystyle (f,\operatorname {CONS} (x,r),s)}. It is said almost, because in both of those results, |s|=|f|−|r|+1{\displaystyle |s|=|f|-|r|+1}. An auxiliary function aux{\displaystyle aux} must then be called for the invariant to be satisfied. Two cases must be considered, depending on whether s{\displaystyle s} is the empty list, in which case |r|=|f|+1{\displaystyle |r|=|f|+1}, or not. The formal definition is aux⁡(f,r,Cons⁡(_,s))=(f,r,s){\displaystyle \operatorname {aux} (f,r,\operatorname {Cons} (\_,s))=(f,r,s)} and aux⁡(f,r,NIL)=(f′,NIL,f′){\displaystyle \operatorname {aux} (f,r,{\text{NIL}})=(f',{\text{NIL}},f')} where f′{\displaystyle f'} is f followed by r reversed.

Let us call reverse⁡(f,r){\displaystyle \operatorname {reverse} (f,r)} the function which returns f followed by r reversed. Let us furthermore assume that |r|=|f|+1{\displaystyle |r|=|f|+1}, since it is the case when this function is called. More precisely, we define a lazy function rotate⁡(f,r,a){\displaystyle \operatorname {rotate} (f,r,a)} which takes as input three list such that |r|=|f|+1{\displaystyle |r|=|f|+1}, and return the concatenation of f, of r reversed and of a. Then reverse⁡(f,r)=rotate⁡(f,r,NIL){\displaystyle \operatorname {reverse} (f,r)=\operatorname {rotate} (f,r,{\text{NIL}})}.
The inductive definition of rotate is rotate⁡(NIL,Cons⁡(y,NIL),a)=Cons⁡(y,a){\displaystyle \operatorname {rotate} ({\text{NIL}},\operatorname {Cons} (y,{\text{NIL}}),a)=\operatorname {Cons} (y,a)} and rotate⁡(CONS⁡(x,f),CONS⁡(y,r),a)=Cons⁡(x,rotate⁡(f,r,CONS⁡(y,a))){\displaystyle \operatorname {rotate} (\operatorname {CONS} (x,f),\operatorname {CONS} (y,r),a)=\operatorname {Cons} (x,\operatorname {rotate} (f,r,\operatorname {CONS} (y,a)))}. Its running time is O(r){\displaystyle O(r)}, but, since lazy evaluation is used, the computation is delayed until the results is forced by the computation.

The list s in the data structure has two purposes. This list serves as a counter for |f|−|r|{\displaystyle |f|-|r|}, indeed, |f|=|r|{\displaystyle |f|=|r|} if and only if s is the empty list. This counter allows us to ensure that the rear is never longer than the front list. Furthermore, using s, which is a tail of f, forces the computation of a part of the (lazy) list f during each tail and insert operation. Therefore, when |f|=|r|{\displaystyle |f|=|r|}, the list f is totally forced. If it was not the case, the internal representation of f could be some append of append of... of append, and forcing would not be a constant time operation anymore.

Note that, without the lazy part of the implementation, the real-time queue would be a non-persistent implementation of queue in O(1){\displaystyle O(1)}amortized time. In this case, the list s can be replaced by the integer |f|−|r|{\displaystyle |f|-|r|}, and the reverse function would be called when s{\displaystyle s} is 0.